.: . -.

||AB =() 0. -

-

---

AB

B ()

A ()

1

1

,

.

,

.

AB

.

,

. , ,

AA .

.

,

, ,..., , ,...u v

AB ,

,

AB

||AB . AB 1, .

;

AB

B ()

A ()

;

;

.: . -.

MA 1

2

AB

AB .

AB

, AB

AB// .

AA

.

AB

,

,

. AB

//AB .

. :

AB

:

)

) .

AB

( )

AB .

B

A

AA

;

;

.: . -.

.

---

1

3

AB

, .

AB

( )

AB .

. AB

,

=AB .

0 .

:

1. ,

MBAM = .

2.

=AB , :

i) =A ( ) ii) =B ( ) iii) =B

;

;

.: . -.

-

.

AB

====0

=

MA 1

4

,

. AB

, :

AB =

= .

:

== ABAB

,

==== .

.

OA =

OB

= .

AOB

, ,

),(

),(

,

, , .

.

00 1800 0 :

====0 ,

.

= ,

.

.

a

a

a

.: . -.

---

AB

B ()

A ()

1

5

=2

,

.

:

, ,

0 . , ,

0 , .

a

.

OA =

AM

= .

OM

+ .

O

AO =

MA

= .

AOOA == /

AAOO = (1)

MAAM

== /

MMAA = (2).

. ( )

1:

.

a

a

a

O

M

+a

+a

.: . -.

MA 1

6

(1),(2) MMOO = , /

MOOM = .

.

. ,

OA =

OB

= ,

+ O O .

. ,

, , , :

(1) a

+=+ ( ) (2) )()(

++=++ ( ) (3) + =0 (4) 0)(

=+ .

:

( )

+ +

+ +( )

+ + ,

,

.

1 2 3, , , ... , , 3 : .

/. ( )

a

a

+

a

a

.

.: . -.

---

AB

B ()

A ()

1

7

1 2 3 1 2 3 1+ + + + = + + + + +... ( ... )

,

1 2 3 4 1 2 3 4+ + + = + + +( ) ( )

,

,...,,, 321 ,

,

.

,

.

:

OMAMOA =+=+

OMBMOB =+=+

.

,

+ = + .

:

OBOBBABOA =+=++=++ )()(

.)()( OAOABABOA =+=++=++

, ( ) ( )

+ + = + +a .

(3) (4) .

2: .

a

+ +

+

a

+ a

a a a

+ +1 2 3

a

1

a a a a

+ + +1 2 3 4

a

2

a

4

a

3

a

a

+

a

a

.: . -.

-

-

.

+ =

=

MA 1

8

.

)(

++++====

,

,

x

, ,

=+ x .

:

=+=++=++=+ xxxx )(0)()()(.

.

,

.

,

,

.

,

OBABOA =+

.

a

a

a

a

a

a

a

+

;

O

.: . -.

---

1

9

OAOBAB =

:

.

. :

)()()(|)()(| ABOAOBABOA + :

| | | | | | | | | | + +

:

1. | | | | | | + = + .

2. | | = + .

0

.

:

( ).

a

+ a

;

.: . -.

,

2,

3

| 3 | = | 3 | 3 | = = |= 3 2 = 6

3

,

6=23|==

||3||3|

MA 1

10

, 0>

, 00

.: . -.

.

---

1

11

||)()(

)()(

=

=

,

||)()(

=

.

ABBA

== .

,

BAAOBO += ,

+=+ )( .

:

:

(i) 00 ======== 0 ==== (ii) )()()( ======== (iii) ==== )( (iv) ==== )( (v) ==== 0 , ==== (vi) ==== 0 , ==== .

a

+

( )a

+

a

a

.: . -.

-

-

. ,

, -

v

523 +=

,

.

0 =

-

=

MA 1

12

.

, ,

53 += ,

32 += .

. ,

v

+= ,

R, .

,

, 0

,

= ,

. . ,

0

,

= . ,

=

| || |

, |||| = . :

,

= | || |

=

,

= | || |

=

.

0

= ,

= 0 .

( iv), , = . :

, , 0

,

//

= , R .

;

.

.: . -.

---

AB

B ()

A ()

1

13

AB .

OM :

AMOAOM +=

BMOBOM += .

,

OBOABMOBAMOAOM +=+++=2 .

2OBOAOM +=

xx ,

OI 1

Ox .

iOI

==== x x . Ox Ox , Ox Ox .

, , xx ,

iOM

// , x

xOM = . x .

,

xOM = x

xx x . )(x .

.

;

x x (x)

.: . -.

Oxy ,

. -

,

x x y y ,

,

i j

MA 1

14

xx y y

i j .

Oxy .

Oxy . y y , xx 1M , xx , y y 2M . x 1M xx y 2M y y , x y . .

.

),( yx ,

: xx )(1 xM y y )(2 yM . 1M 2M

y y xx , . .

x y M x y( , ) ( , )x y .

.

3: jyix ++++==== .

;

i

j

x

y

y

x

1

2 (x,y)

.: . -.

---

AB

B ()

A ()

1

15

: Oxy .

OA = . 1A 2A xx y y , :

21 OAOAOA += (1)

yx, A ,

xOA =1 jyOA =2 . (1) jyix ++++====

i

j .

: x y .

i

j . , jyix += . jyixjyix +=+ jyyixx )()( = xx , 0 xx ,

i y yx x

j=

, ,

i j/ / , , i

j . xx = , yy = . :

jyix ++++==== .

xi

yj

i

j , yx, Oxy .

a

a

i

j

A1

y

x

A 2

.: . -.

-

-

.

,

)1,1( = )2,1(=

,

:

(1, 1) (1,2) + = + =(2,1)= ,

)2,2()1,1(22 == ,

(1, 1) (1,2) = =(1, 1) ( 1, 2) (0, 3)= + =

MA 1

16

, x

y

.

( , )x y .

,

+ ,

, R

. ,

),( 11 yx=

),( 22 yx=

, :

+ = + + + = + + +( ) ( ) ( ) ( )x i y j x i y j x x i y y j1 1 2 2 1 2 1 2 jyixjyix )()()( 1111 +=+=

+ = + +( , )x x y y1 2 1 2

= ( , )x y1 1

),(),(),( 21212211 yyxxyxyx ++=+

),(),( 1111 yxyx =

,

+ :

),(),(),( 21212211 yyxxyxyx ++=+=+

.

),( 11 yx ),( 22 yx ),( yx .

.

.

.: . -.

---

1

17

)(

21 OBOAOM +=

),( yxOM = , ),( 11 yxOA= , ),( 22 yxOB=

: )],(),[(21),( 2211 yxyxyx +=

++=

2,

22121 yyxx

221 xxx

+=

221 yyy

+= .

),( 11 yx ),( 22 yx ),( yx

AB .

:

OAOBAB = , ),( yxAB= , ),( 22 yxOB=

),( 11 yxOA= : ),(),(),(),( 12121122 yyxxyxyxyx == . :

),( yx ),( 11 yxA ),( 22 yx

12 xxx = 12 yyy = .

:

=AB -

=AB - .

, AB )2,1(

)7,3( x = =3 1 2 527 ==y , )5,2(= .

y

x

A(x1,y1)

B(x2,y2)

(x,y)

.

y

x

A(x1,y1)

B(x2,y2)

.: . -.

MA 1

18

),( yx=

OA = . 1 2 xx yy , x y , ||)( 1 x= ||)( 2 y= . :

222222

21

21

21

22 ||||)()()()()(|| yxyx +=+=+=+== :

),( yx= , 22|| yx +=

, )12,5(= , 13125|| 22 =+= .

( , )x y1 1 ),( 22 yx . )(

),( 1212 yyxxAB = ,

22|| yx += :

212

212 )()()( yyxx +=

:

( , )x y1 1 ),( 22 yx

212

212 )()()( yyxx += .

.

.

a

A1

y

x

A(x,y) 2

y

x

A(x1,y1)

B(x2,y2)

.: . -.

---

1

19

= ( , )x y A

aOA = . , Ox ,

x x . : 20

.: . -.

= ( , )3 1

= ( , )3 3 -,

det( , )

=

=

3 13 3

3 3 0= + = ,

= ( , )2 3 ,

= ( , )1 2 ,

det( , )

=

= +2 31 2

4 3

3 7 0+ = .

MA 1

20

),( 11 yx=

),( 22 yx=

.

:

1 1

2 2

// 0x yx y

=

x yx y

1 1

2 2, 1

2

,

( ) ),det(

. ,

:

// det( , ) 0a =

= ( , )x y1 1

= ( , )x y2 2 1 2 . :

1 1 1 21 2 2 1 1 2

2 2 1 2

// 0x y y y

x y x yx y x x

= = = =

.

,

1 2 :

1 2// =

.

.: . -.

22 : x yx y

1 1

2 2= 1 2 1 2x y y x

det( , )a =

x yx y

1 1

2 2

---

1

21

),( 11 yx=

),( 22 yx=

.

0

, R , ,

= . ,

),(),( 2211 yxyx = : 21 xx = 21 yy = ,

x y y x x y y x1 2 1 2 2 2 2 2 0 = =

022

11 =yxyx

.

0

= , x yx y

x y1 12 2

1 1

0 00= = .

,

: 022

11 =yxyx

.

, 022

11 =yxyx

,

. , 022

11 =yxyx

,

1221 yxyx = . ,

02 x , 22

11 y

x

xy = , , =

2

1x

x,

: 21 xx = 21 yy = .

,

=

// .

02 =x , 021 =yx , 01 =x ,

,

, , 02 =y ,

,

.

1 1

2 2

x y// 0

x y = = = =

.: . -.

,

| | = 3 , | | = 8

3

=

122183

383 ===

-

-

.

MA 1

22

| | | | = ,

.

= 0

= 0 ,

= 0

: :

==== ( )

, 0====

.

, |||| ==== .

, |||| ==== .

2

. : 22 ||0|||| == . 2 2| |==== .

,

i j

: 0======== ijji

122 ======== ji

= ( , )x y1 1

= ( , )x y2 2 . :

1 2 1 2x x y y = +

;

.

A

F

.: . -.

---

1

23

= ( , )x y1 1

= ( , )x y2 2 .

OA =

OB

= .

+= ))((2)()()( 222 ,

,,

. :

( ) ( ) ( ) 2 2 1 2 2 1 2= + x x y y , ( ) 2 12 12= +x y ( ) 2 22 22= +x y . , :

+++=+ ))((2)()( 22222121212212 yxyxyyxx

+++=+++ ))((222 22222121212221212221 yxyxyyyyxxxx

= ))(( , :

1 2 1 2x x y y = +

,

= ( , )x y1 1 ,

= ( , )x y2 2

= ( , )x y3 3 , : (1) 1 1 2 2 1 2 1 2 1 2 1 2( ) ( , )( , ) ( ) ( ) ( )x y x y x x y y x x y y = = + = + =

( ) =

1 1 2 2 1 2 1 2 1 2 1 2( ) ( , )( , ) ( ) ( ) ( )x y x y x x y y x x y y = = + = + =

( ) =

: ( ) ( ) ( )

= =

1 2 1 2 = + = + = + = +

x x y y

:

(1) ( ) ( ), R = = = = = = = =

(2) ( ) + = + + = + + = + + = +

(3) 1 2 1 = = = =

a

y

x

(x1,y1)

(x2,y2)

.: . -.

= ( , )1 2

= ( , )31 , :

2 2 2 2

1 3 2 1

1 2 3 1

+ = =

+ +

5 1 225 10 2

= = =

=4

.

-

.

MA 1

24

(2) + = + + = + + +( ) ( , )( , ) ( ) ( )x y x x y y x x x y y y1 1 2 3 2 3 1 2 3 1 2 3

= + + + = + + +( ) ( ) ( ) ( )x x x x y y y y x x y y x x y y1 2 1 3 1 2 1 3 1 2 1 2 1 3 1 3= +

(3) 1 2 1 2 1 2 1 20 0x x y y y y x x = + = =

1 21 2

1 2

1 1y yx x

= =

= ( , )x y1 1

= ( , )x y2 2 ,

=| | | | , ||||

==== .

,

= +x x y y1 2 1 2 , | | = +x y12 12 | |

= +x y22

22

.

: 22

22

21

21

2121yxyx

yyxx

++++++++

++++====

v

, 0

.

OA =

OM = .

OA 1M .

1OM

. :

=1OM

.

:

==+=+= OMMMOMMMOMv 11111 )(

: =

.

.

a

M1

M

A v